Last Digits
Use shortcuts to find just the last digit of each answer – there's no need to calculate the rest!
This course starts at the very beginning — covering all of the essential tools and concepts in number theory, and then applying them to computational art, cryptography (code-breaking), challenging logic puzzles, understanding infinity, and more!
In these warmups, if you know the trick, you'll finish each problem in seconds!
Use shortcuts to find just the last digit of each answer – there's no need to calculate the rest!
Look for patterns, and when you think you've found one, use it to decode the message!
Investigate the coloring rules that apply when you do math on a rainbow-striped number grid.
Every integer greater than 1 has a unique name that can be written down in primes.
How much can you learn about a number if you can only see its last digit?
Review the divisibility shortcuts that apply when you're dividing by a power of 2 or 5.
Explore the pattern of what remainders remain when you divide powers of 10 by 9 or 3.
Apply divisibility rules as well your own logic to determine just the last digit of each solution.
How do the remainders of an operation's inputs impact the remainder of the calculation output?
Investigate surprising patterns that surface when you calculate digital roots.
Factor each number one step at a time until every piece that you have is a prime.
Learn to use factorization as a versatile problem-solving tool.
Since they're defined as products, what happens when you factor them?
Learn a quick technique for determining how many different divisors a number has.
Imagine a long hallway with 100 closed doors numbered 1 to 100...
Are there finitely many prime numbers or infinitely many of them, and how can you be sure either way?
Explore, apply, and relate the GCD and LCM.
Again, imagine that long hallway of doors, but this time focus your attention on exactly who does what.
Build intuition for where least common multiples appear in both abstract and real-life contexts.
Explore how the path of a ball bouncing around a pool table is affected by the table's dimensions.
Use prime factorization as a tool for finding the greatest common divisors of pairs of numbers.
When you draw a right triangle on a grid of dots, how many dots does does the hypotenuse cut through?
When do these jumping rules allow you to reach every number on the number line?
Develop a systematic procedure to determine the smallest positive integer that you can reach by jumps.
What's the pattern to these answers? What's going on in the big picture here?
Understand how the GCD and LCM are related by thinking about factors arranged in a Venn diagram.
Explore the patterns created when pool balls "paint" the squares they touch as they roll.
How do you get back "home" to the bottom left pocket?
The danger of cyclic systems: one step too far and you're back where you started!
Time, as measured by a clock or calendar, is "modular," so let's start there...
What happens when you wrap an infinite number line around a one-unit square?
Learn and practice doing arithmetic in the modular world.
Review the rules for arithmetic with remainders and uncover the peculiar divisibility rule for 11.
You probably know how to draw a 5-pointed star, but what about an 8 or 12 or 30-pointed star?
Learn a general formula for the number of points a star will have.
How many different 60-point stars can you make, drawing just one path on a circle of 60 points?
The challenge is to measure out a specific quantity of liquid using only a few types of legal moves.
Which pairs of containers can measure out any quantity of liquid and which ones have limited use?
Considering the remainder "modulo" an integer is a powerful tool with many applications!
Explore a concept that's lurking beneath the surface of both star drawing and decanting puzzles.
Can normal equations with no integer solutions be converted into congruences that DO have solutions?
What does exponentiation look like in modular arithmetic?
Use the factorization of a number to determine how many small numbers are relatively prime to it.
This isn't Fermat's fearsome Last Theorem, but it still packs a big punch for a little guy!
Use Euler's theorem to quickly find just the last few digits of enormous exponential towers!
Leverage what you know about totients to find any card in the deck after a series of perfect shuffles.
Explore one of the most misunderstood concepts in math - infinity.
To understand cardinal infinity, first start by counting and comparing finite sets.
Explore a crazy world of numbers that contains infinitely many infinities, both small and large.
Help Hilbert use his hotel that has infinitely many rooms to host infinitely many sleepy guests.
Can a betting game have an infinite expected value?